New Method for Homogeneous Smoluchowski Coagulation Equation

In this paper, variational iteration method is employed to solve the homogenous Smoluchowski coagulation 
equation. The intervals of validity of the solutions will be extended by using Pade approximation. Error will be decrease, as it is expected. The numerical results show the effectiveness and the simplicity of the methods.


Introduction
Variation iteration method (VIM) has been proposed by Ji-Huan He, in 1998, and has been applied to solve many linear and nonlinear functional equations. There are many research documents, in the literature, for application of VIM to solve different functional equations, such as [1][2][3].
Let's consider the following non-linear functional equation, Where L, N and g(t) are a linear, a non-linear operator, and a known analytic function, respectively. In this method, a correction functional including a general Lagrange multiplier, will be constructed as follows, Lagrange multiplier can be identify optimally via the variation theory. An iterative formula, for computing the sequence of the approximations, will be obtained as soon as the Lagrange multiplier is determined. The successive approximations ( ), 0, ≥ n u t n of ( ) u t will be obtained by selection an initial approximation of the solution, u 0 . Initial and boundary conditions must be satisfied by initial approximation, u 0 . Iterative formula is constructed as follows; Exact solution will be determined as the following limit,

Pade Approximation
The series solution obtained by VIM has a small region of convergence. To extend the region of convergence, Pade approximation will be helpful. Pade approximation of a function is given by the ratio of two polynomials [4]. The coefficients of the polynomials in the numerator and denominator are determined by using the coefficients in the Taylor series expansion of the function. The Pade approximation of a function is shown as the following, Where c i 's are known coefficients and, a i 's and b i 's should be determined. The numerator and denominator have no factors in common.

Homogeneous Smoluchowski Coagulation Equation
The physical process of coagulation of particles is often modeled by Smoluchowski's equation. This equation is widely applied to describe the time evolution of the cluster-size distribution during aggregation processes. In this paper, the following Smoluchowski's equation [5,6], will be considered; ( , ) ( ) ( ), , , Where u 0, is a known function. u(x, t), is the density of cluster of mass x per unit volume at time t.

Applications
To illustrate the ability and the simplicity of the methods two examples are presented.
Example 1: Equations. (6)-(9) are considered with constant kernel, k (x, y)=1, and u 0 =exp (-x), [5][6][7] 0 0 x u x t u x y t u y t dy u y t u x t dy t (10) The exact solution is, 2 ( , ) ( )exp( ( ) ), , , where λ is general Lagrange multiplier,  n u is considered as restricted variations. To determine the value of λ the following procedure should be followed, which is equivalent to, Other approximations easily obtain by (17 Exact solution, approximate solutions via VIM, and VIM-Pade, and absolute errors of these two methods are plotted in Figures 1a-1k. x u x t x y y u x y t u y t dy xy u y t u x t dy t with the exact solution, And I 1 is the modified Bessel function of the first kind,